Problem: Simplify the following expression: $y = \dfrac{-40z^2 + 40z}{8z^2 + 40z}$ You can assume $z \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-40z^2 + 40z = - (2\cdot2\cdot2\cdot5 \cdot z \cdot z) + (2\cdot2\cdot2\cdot5 \cdot z)$ The denominator can be factored: $8z^2 + 40z = (2\cdot2\cdot2 \cdot z \cdot z) + (2\cdot2\cdot2\cdot5 \cdot z)$ The greatest common factor of all the terms is $8z$ Factoring out $8z$ gives us: $y = \dfrac{(8z)(-5z + 5)}{(8z)(z + 5)}$ Dividing both the numerator and denominator by $8z$ gives: $y = \dfrac{-5z + 5}{z + 5}$